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Theorem bj-1uplex 33487
Description: A monuple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-1uplex (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)

Proof of Theorem bj-1uplex
StepHypRef Expression
1 bj-pr11val 33484 . . 3 pr1𝐴⦆ = 𝐴
2 bj-pr1ex 33485 . . 3 (⦅𝐴⦆ ∈ V → pr1𝐴⦆ ∈ V)
31, 2syl5eqelr 2884 . 2 (⦅𝐴⦆ ∈ V → 𝐴 ∈ V)
4 df-bj-1upl 33477 . . 3 𝐴⦆ = ({∅} × tag 𝐴)
5 p0ex 5054 . . . 4 {∅} ∈ V
6 bj-xtagex 33468 . . . 4 ({∅} ∈ V → (𝐴 ∈ V → ({∅} × tag 𝐴) ∈ V))
75, 6ax-mp 5 . . 3 (𝐴 ∈ V → ({∅} × tag 𝐴) ∈ V)
84, 7syl5eqel 2883 . 2 (𝐴 ∈ V → ⦅𝐴⦆ ∈ V)
93, 8impbii 201 1 (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wcel 2157  Vcvv 3386  c0 4116  {csn 4369   × cxp 5311  tag bj-ctag 33453  bj-c1upl 33476  pr1 bj-cpr1 33479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-nel 3076  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-xp 5319  df-rel 5320  df-cnv 5321  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-bj-sngl 33445  df-bj-tag 33454  df-bj-proj 33470  df-bj-1upl 33477  df-bj-pr1 33480
This theorem is referenced by:  bj-2uplex  33501
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